Abstract

In this work we introduce a tunable GMR filter based on continuously period-chirped (ΔP = 130 nm) gratings using a Ta2O5 waveguide layer with graded thickness (ΔT = 36 nm). The structure of the gradient-period grating is defined using a modified Lloyd’s mirror interferometer with a convex mirror, and Ta2O5 film used for the gradient is deposited using masked e-beam evaporation. The as-realized chirped GMR filter provides sharp transmission dips at resonant wavelengths with a filter bandwidth of approximately 4.2 nm and 0.78 nm when respectively applied to TE and TM polarized light under normal incidence. Gradually sweeping the chirped GMR filter makes it possible to monotonically sweep through resonant wavelengths from 500 to 700 nm, while maintaining stable filter bandwidth and transmission intensity. The optical spectrum of the incoming light can then be loyally reconstructed accordingly. We successfully demonstrate the spectrum reconstruction of a white light emitting diode and a dual-peak laser beam using the proposed chirped GMR filter as a dispersive device.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Conventional spectrometers based on diffraction gratings or prisms provide high spectral resolution by increasing the distance to the detection element to facilitate discrimination among wavelengths. However, these devices are too bulky for integration with handheld devices, such as smartphones. Several on-chip spectroscopy schemes have been proposed, including arrayed waveguide gratings [1–3], echelle diffraction gratings [4,5], super-prism-based photonic crystals [6], micro-resonator arrays [7], and filter arrays based on Fabry-Perot cavities [8,9]. However, these techniques necessitate a compromise between spectral resolution and operating bandwidth, thereby limiting their practical applicability. Guided-mode resonance (GMR) filters associated with the diffraction anomaly of periodic surface structures are a unique class of narrow-band filters with nearly 100% reflection efficiency [10]. The physics behind GMR filters is based on the excitation of resonant leaky Bloch modes propagating in the periodic lattice [11]. GMR filters can be used as the dispersive element in compact spectrometer systems; however, the filter response must be narrowband and wavelength-tunable. The wavelength of GMR filters can be tuned by harnessing the electro-optic effect, thermo-optic effect, or by employing microelectromechanical systems or opto-fluidics; however, all of those schemes provide wavelength tuning over a narrow range (< 30 nm) [12–14]. We recently demonstrated that the resonant wavelength of a GMR filter can be selected by controlling the angle of incidence, and the corresponding reflected color can be shifted from near-infrared (710 nm) to blue (430 nm) [15]. Unfortunately, this approach also requires wide spacing between the GMR filter and detection element to achieve wavelength discrimination. GMR filters with a gradient-period grating (also referred to as chirped GMR filters) also provide wide-range wavelength tuning. These devices exhibit spatially-dependent resonance that is excited via phase-matching at a specific location on the grating structure. This means that calibrating the chirping profile of the GMR filter to extract the grating period along the direction of interference enables reconstruction of the optical spectrum of the incident light by sweeping the light beam across the chirped GMR filter. Alternatively, an optical detector array can be positioned behind the chirped GMR filter to achieve rapid detection without the need to sweep the incident light or GMR filter.

The fabrication of chirped grating structures typically requires expensive, time-consuming e-beam lithography or focused ion-beam lithography [16–19]. As-realized chirped gratings are characterized by a stair-step approximation using a grating period graded in finite increments. This allows chirped GMR filters to exhibit a broader spectral linewidth (55 nm for TE incidence and 10~12 nm for TM incidence); however, they provide poor spectral resolution when used for wavelength discrimination [18]. Continuously-chirped GMR filters can also be fabricated by casting a stretched wedge-shaped PDMS grating [20] or by employing a waveguide layer with graded thickness [21]. This type of GMR filter enables a finer spectral linewidth (down to only 2.7 nm for TM incidence); however, a limited wavelength detection range (Δλ < 50 nm) makes it difficult to cover the entire visible spectral region, which is required for most practical applications.

Laser interference lithography (LIL) is an effective technique for patterning regular arrays of fine features over a large sample area without the need for complex optical systems or photomasks. Wafer-scale nano-patterning of grating structures with constant, uniform periodicity has been achieved using a one-beam Lloyd’s interferometer [22] as well as a two-beam interferometer [23]. However, chirped grating patterns with progressively varying periods require the insertion of a refractive optical element, such as a cylindrical lens or concave mirror, in the LIL system to vary the incident angle of one of the two interfering laser beams, in order to produce true continuously-chirped gratings [24–28].

In this work, we demonstrate the first continuously-chirped GMR filter defined using a one-beam Lloyd’s interferometer equipped with a convex mirror. Compared to a concave mirror with the same curvature [28], a convex mirror allows for a more gradual gradation in incident angle and larger interference area. This is particularly advantageous for the spatially-resolved chirped GMR filters used in high-resolution on-chip spectroscopy. Finally, we demonstrate a chirped GMR filter using both the graded grating period and the graded thickness of Ta2O5 waveguide to enable constant transmission across a wide wavelength range (exceeding 200 nm), which cannot be achieved using only gratings with graded period. The proposed device also provides superior spectral resolution of 0.78 nm for TM polarization.

2. Design and experiments

2.1 Chirped GMR filter simulation

Figure 1 presents a schematic diagram showing a continuously-chirped GMR filter comprising a low-index chirped grating pattern covered with a high-index Ta2O5 overlay. When the chirped GMR filter is illuminated by a monochromatic light source, the incident light is reflected only when applied at a specific location where the phase matching condition is valid. Otherwise, the light simply passes through the chirped GMR filter. Conversely, illuminating the chirped GMR filter via a white light source results in the formation of a rainbow-colored reflection image. Figure 2(a) shows the simulated optical transmission spectra of the proposed chirped GMR filter with a 110-nm-thick Ta2O5 layer in which the incident light applied to the chirped GMR filter is TE polarized. Figure 2(b) presents the same results in which the incident light is TM polarized. The simulations are based on rigorous coupled-wave analysis (RCWA), while taking into account the dispersion and absorption of the constituent materials. The simulation model was set up in accordance with the schematic diagram in Fig. 1. In the simulation model, the thickness of the grating structure is 110 nm and the duty cycle is 0.4. The period of the gradient grating monotonically varies from 250 nm to 400 nm, resulting in a gradient resonant wavelength ranging from 430 nm to 650 nm. We assumed that the surrounding environment is air with the refractive index set to unity. In this case, the proposed chirped GMR filter provides wideband (> 10 nm) optical reflections of TE-polarized light, while simultaneously enabling narrowband optical reflection of TM-polarized light, as shown in Figs. 2(a) and 2(b). Figure 2(a) reveals the existence of higher-order TE optical reflections at wavelengths shorter than the resonant wavelength, which would deteriorate filter performance in spectroscopic applications. This issue can be mitigated by reducing contrast in the refractive index between the Ta2O5 waveguide layer and the surrounding environment through the use of a refractive index liquid, as shown in Fig. 2(c). This would also enable a narrower reflection bandwidth for TE polarization, compared to the results in Fig. 2(a). Figure 2(c) shows that the proposed chirped GMR filter operates across a limited range of wavelengths. In this case, the sharp dip in transmission shrinks and vanishes for wavelengths longer than 650 nm. This can be attributed primarily to the fixed thickness of the Ta2O5 waveguide layer, which efficiently couples specific incident waves to a leaky waveguide mode via phase matching, whereupon the GMR filter reradiates the light in the form of normal optical reflectance [29]. Realizing a chirped GMR filter with phase matching covering a wider wavelength range requires a Ta2O5 waveguide with graded thickness, as shown in Fig. 2(d). In this case, the ratio of the thickness of the Ta2O5 waveguide to the grating period is set at 0.63.

 

Fig. 1 Schematic diagram showing continuously-chirped Ta2O5 guided mode resonance filter. Gradient variation in the grating period is crucial to creation of a compact high-resolution optical spectrometer.

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Fig. 2 (a,b) Simulated optical transmission spectra of continuously-chirped air-cladded Ta2O5 based GMR filter under normal incidence TE- and TM-polarized light, respectively. (c,d) Simulated optical transmission spectra of chirped GMR filter filled with refractive index liquid for TE polarization, as well as a Ta2O5 waveguide layer with (c) fixed and (d) graded thicknesses.

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2.2 Concept of convex Lloyd’s mirror

A continuously-chirped grating structure was obtained using a one-beam Lloyd’s laser interferometer with modified configuration aimed at varying the incident angle of one of the two interfering laser beams, as shown in Fig. 3(a). In this LIL setup, a cylindrically convex mirror with a nominal curvature radius R of 193 mm was used to replace the planar mirror found in conventional Lloyd’s interferometers. The divergence in the laser beam leaving the convex mirror produces a progressively varying incident angle θ2, whereas incident angle θ1 (for the direct laser beam) remains constant regardless of the sample position. This results in a continuously period-chirped grating structure, the minimum period range of which is defined by the incident angle of the direct laser beam. Thus, the period of the resulting grating pattern on the sample is given by:

Λ=λuvsinθ1+sinθ2=λuvsinθ1+sin[θ1+2sin1(x/R)]
where λuv is the wavelength of the laser source, θ1 is the incident angle of the direct laser beam, R is the nominal curvature radius of the convex mirror, and x is an arbitrary position on the x-axis, where the light is reflected from the concave mirror and subsequently arrives at position y on the sample. Physical parameters x0 and x1, which are defined in Fig. 3(a), refer to the distances between the convex mirror and the Lloyd’s intersection point and half diameter of the convex mirror, respectively. The relationship between x and φ is given by:
φ=2sin1(x/R)
where a special case of φ = 0 occurs at the center position of the convex mirror (x = x0 + x1), for which the reference coordinates y0 and y1 and subsequently the y-coordinate are defined as follows:

 

Fig. 3 (a) Schematic illustration of one-beam Lloyd’s laser interferometer equipped with a convex mirror. Physical parameters x0, x1, y0, y1, and R are graphically defined in the figure. (b) Chirped grating periods as a function of interference position y calculated for convex Lloyd’s mirrors with three different curvature radii: R = 193, 500, and 1000 mm.

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y(φ)=(Rsinφ+x0+x1)tan(θ1+2φ)y1y0y1=(x0+x1)tanθ1y0=R(1cosφ)

This means that the grating period and the corresponding y-coordinate can be calculated as functions of φ using Eq. (1-3). In Fig. 3(b), the calculated chirped grating periods are plotted as functions of interference position y for convex Lloyd’s mirrors with three different curvature radii. In these calculations we assume that λuv = 355 nm, x0 = 19 mm, x1 = 34 mm, and θ1 = 45° based on the actual LIL system fabricated in this study. Simulation results demonstrate that a highly-curved convex Lloyd’s mirror allows strong dispersion in the grating period over a large interference area. Concave Lloyd’s mirrors with the same curvature radius provide similar dispersion behavior in terms of grating period; however, this type of grating is implemented within a smaller interference area. For high-resolution and wide-range spectroscopy applications, it is preferable to employ chirped gratings using finely-graded grating periods covering a large sample area. This is the approach selected for the proposed convex mirror LIL.

2.3 LIL system and chirped GMR filter fabrication

One of the roadblocks to achieving uniform grating patterns over a large sample area is the Gaussian intensity distribution inherent to laser beams. This causes the exposure dose in a Lloyd’s mirror interferometer to vary across the exposure area, with the maximum exposure intensity occurring at the intersection between the mirror and the sample stage [22,23], resulting in variations in the duty cycle of the resulting grating pattern. In the case of a chirped grating definition using a convex Lloyd’s mirror interferometer, the longest grating period occurs at the intersection point of the Lloyd’s stage with a gradual decrease in the grating period in the direction away from the intersection point. This means that when using a Gaussian laser light field, the exposure energy is lower in the regions where short-period gratings are formed. This contradicts the LIL rule of thumb that the formation of short-period gratings requires a higher exposure energy. This should result in a larger variation in the duty cycle of the resulting chirped grating. We employed a flat-top laser light field in a Lloyd’s laser interferometer system in the fabrication the chirped grating, as shown in Fig. 4. This LIL system comprises a highly-coherent 355-nm diode-pumped solid-state laser (Cobolt ZoukTM), a volume-grating-based spatial filter, a variable optical attenuator based on two polarizing cube beam splitters with half-wave plate, a beam shaping device consisting of a tunable beam expander and a refractive beam shaper, a beam collimator, and a Lloyd’s sample stage equipped with a convex mirror. The laser has an output power of 20 mW and a spectral linewidth less than 1 MHz. Details pertaining to the operating principle underlying beam shaping in this LIL system can be found in our previous publication [22]. In this upgraded system, we sought to ensure the collimation of the expanded laser light field through the use of a 6-inch plano convex lens, thereby enabling improved control over the incident angle of light transmitted onto the convex mirror and sample. This enabled fabrication of chirped gratings with a precisely defined period. The convex mirror used in this study had a radius of curvature of 192 mm, a scratch-dig surface quality of 20-10, a surface figure of less than λ/4 at λ = 633 nm, and a high-reflection multilayer dielectric coating tuned for 355 nm.

 

Fig. 4 Schematic illustration of proposed LIL system used for chirped grating fabrication.

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Figure 5 presents the grating period versus interference position under a direct laser beam at various incident angles θ1. The solid lines in the figure indicate theoretical predictions based on Eqs. (1-3), whereas the hollow symbols indicate experimental data obtained using an atomic force microscope (AFM). In the proposed LIL setup, the range of the gradient grating period was defined primarily by incident angle θ1 of the direct laser beam. When the incident angle was 20 degrees, the resulting grating period varied from 400 nm to 800 nm, covering 400 nm of grating period range over a sample area of 5 x 5 cm. The insets of Fig. 5 are AFM images of the grating structures at interference positions of 5, 20, and 50 mm, respectively. Figures 1 and 3 show that the grating structures had a constant thickness of 60 nm at every point of interference along the y-axis. The variation in duty cycle is only 5%. Even when using a flat-top light field, we observed a slight increase in the duty cycle at short-period gratings, thereby confirming the aforementioned rule of thumb. When the incident angle was increased to 60 degrees, the range of the gradient-period grating was reduced to only 25 nm (215 to 240 nm) across the entire 5 x 5 cm sample area. Figure 5 and Eq. (1) show that the relationship between the grating period and interference position is not a linear function. This issue can be resolved by replacing the cylindrically convex mirror with an aspheric lens of a varying curvature radius. In this proof-of-concept demonstration photoresist grating pattern is employed to realize the chirped GMR filter; however, we can further transfer the grating patterns into dielectric material for a robust device in practical applications.

 

Fig. 5 Grating period versus interference position using direct laser beams with various incident angles θ1. Solid lines indicate theoretical predictions, whereas hollow symbols indicate experimental data.

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The grating parameters (period, thickness, and duty cycle) can be extracted using an atomic force microscope; however, this would allow for inspection of only a small area and this would prove time-consuming. Optical diffractometers are widely used to characterize grating structures, particularly to determine the grating period [30–32]. We had developed an automated optical diffractometer system to provide the rapid wafer-scale mapping of periodicity and diffraction efficiency of grating structures. This optical diffractometer system utilizes the same 355-nm laser of LIL system as the light source, which illuminates the sample with an incident angle of 65 degree. The polarization of probing laser is parallel to the grating stripes. The setup of this system will be detailed elsewhere. We employed this optical diffractometer system to characterize the chirped gratings defined on a 2-inch silicon wafer. This system helped to resolve a continuously-chirped grating pattern with periodicity monotonically varying from 350 nm to 688 nm, as shown in Fig. 6. Diffraction efficiency is determined primarily by the thickness and duty cycle of the grating; therefore, the mapping of diffraction efficiency can be used to reveal the uniformity of chirped gratings. The slight reduction in diffraction efficiency at longer and shorter grating periods can be respectively attributed to the smaller and larger duty cycle of the gratings.

 

Fig. 6 Mapping of (a) periodicity and (b) diffraction efficiency of chirped gratings over 2-inch wafer area using custom-made optical diffractometer.

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According to the RCWA simulations in Fig. 2, it can be difficult to maintain the transmission bandwidth and intensity of chirped GMR filter over a wide wavelength range when the waveguide layer has a constant thickness across the sample area. The red rectangles in Fig. 7 indicate the optimal thickness range for a waveguide layer with a given grating period applied to TE polarization. A GMR filter with a waveguide thickness within this optimal range should theoretically provide sharp transmission at its resonant wavelength with zero transmission intensity. This means that a chirped GMR filter with a fixed Ta2O5 thickness of 110 nm would be able to provide high-extinction-ratio spectral responses only in short wavelength regions, as indicated by the blue triangles in Fig. 7. The thickness of the graded Ta2O5 waveguide layer demonstrated in this work (green circles) generally follows optimal thickness trends (red rectangles) for chirped GMR filters, which should result in constant transmission bandwidth and intensity over a wider wavelength range.

 

Fig. 7 Cross-sectional SEM views of chirped GMR filter in positions where grating period is (a) 277 nm and (b) 394 nm; (c) thickness trends of Ta2O5 waveguide layer in two as-fabricated chirped GMR filters (red rectangles indicate optimal thickness range for given grating period, based on RCWA simulations).

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3. Results and discussions

In this work, we characterize the chirped GMR filters using three approaches. First, we illuminated the as-realized chirped GMR filter at a set angle using a collimated white light beam from a solar simulator in order to visually check its rainbow-colored diffraction image, as shown in Fig. 8(a). We then transferred the sample to a polarizing microscope to check its GMR performance based on optical reflection images. Second, we inserted the chirped GMR filter into an optical system in which a collimated and polarized white light beam was used to illuminate the sample under normal incidence, with an optical fiber serving as a receiving aperture to collect and guide the transmitted light to a commercial optical spectrometer (USB4000 from Ocean Optics) with optical spectral resolution of 1~2 nm. There was the possibility that an optical spectrometer with such limited resolution would prevent us from observing the filter response for TM polarization, which generally has a linewidth of less than 1 nm, as shown in Fig. 2. To resolve this issue and extract the intrinsic spectral linewidth of the chirped GMR filter, we inserted the sample into the optical system shown in Fig. 8(b). In this system, the white light source was replaced with red and green laser sources, and the optical spectrometer was replaced with an optical power meter. The linewidth of the laser beam used in this experiment was finer than 0.1 nm. This made it possible to extract the intrinsic spectral linewidth of the chirped GMR filter as follows: (1) the laser beam was swept across the sample along the direction of interference, (2) the energy of the transmitted light was detected at every position, (3) the interference position was converted to the grating period using Eq. (1) or the experimental results in Fig. 5, and (4) the grating period was converted to the wavelength domain in accordance with RCWA simulations (linear relationship). The sweeping of laser beam was achieved by laterally moving the sample holder via a high-precision motorized linear stage (UTS 150CC from Newport), while keeping the laser source and receiving fiber motionless.

 

Fig. 8 (a) Photograph of chirped GMR filter illuminated by white light source to reveal rainbow-colored diffraction image; (b) experiment setup used to characterize spectral resolution of chirped GMR filter.

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Figures 9(a) and 10(a) present the measured optical transmission spectra of as-realized chirped GMR filters versus the interference position for TE polarization using Ta2O5 waveguides with a fixed or graded thickness, respectively. These experiments were based on the aforementioned second approach using a white light source and a commercial optical spectrometer with a light beam sweeping at intervals of 200 μm. The corresponding transmission bandwidth and intensity at all interference positions (equivalent to all resonant wavelengths of the filter) are summarized in Figs. 9(b) and 10(b), respectively. The transmission bandwidth refers to the spectral response linewidth at absolute 0.5 optical intensity in the transmission spectrum. The chirped GMR filters were filled with refractive index liquid to suppress higher order resonance and further reduce filter linewidth. The experiment results in Fig. 9 reveal sharp transmission dips at the corresponding resonance wavelengths of the chirped GMR filter when using a Ta2O5 waveguide with thickness of 110 nm. The refractive index liquid used for this sample had a refractive index of 1.7. In this case, the widest transmission bandwidth of approximately 6.5 nm occurred at a wavelength of 500 nm. For wavelengths longer and shorter than 500 nm, we observed an increase in transmission intensity at wavelengths from 0.15 to 0.4, indicating that the extinction ratios at both longer and shorter wavelengths were degraded. As a result, the transmission bandwidth varied from 3 to 6.5 nm across the entire 200-nm wavelength range. These experiment results are in good agreement with the theoretical predictions shown in Fig. 2(c). By employing a Ta2O5 waveguide with graded thickness in the chirped GMR filter, we were able to achieve a stable filter linewidth of approximately 4.5 nm and transmission intensity of 0.18 across the 200-nm wavelength range, as shown in Fig. 10(b). The higher transmission losses at shorter non-resonant wavelengths (below 520 nm) in Fig. 10(a) can be attributed to absorption from the refractive index liquid (n = 1.8), which was not an issue in Fig. 10(a) due to the high transparency of the refractive index liquid in that experiment (n = 1.7).

 

Fig. 9 Chirped GMR filter with a fixed waveguide layer: (a) Measured optical transmission spectra of chirped GMR filter and (b) corresponding transmission bandwidths and intensities along direction of interference for TE polarization (thickness of Ta2O5 waveguide layer fixed at 110 nm).

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Fig. 10 Chirped GMR filter with a graded waveguide layer: (a) Measured optical transmission spectra of chirped GMR filter and (b) corresponding transmission bandwidths and intensities along direction of interference for TE polarization (thickness of Ta2O5 waveguide layer 126~162 nm).

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In addition, we employed the aforementioned third approach to further examine the spectral resolution of the gradient-thickness chirped GMR filter for both TE- and TM-polarized light. During this experiment, we utilized a fine laser beam movement interval of 20 μm to ensure high spectral resolution in the reconstructed spectrum. The experiment results in Fig. 11 reveal the importance of refractive index liquid in reducing the index contrast, and thereby reducing the filter linewidth, particularly at longer wavelengths. The ultimate spectral resolution of the as-realized gradient-thickness chirped GMR filter was approximately 4.2 nm and 0.78 nm for TE and TM polarization, respectively. The refractive index liquid used in chirped GMR filter can be replaced with spin-on glass coatings to avoid the liquid. Further reducing the filter linewidth can be possible by optimizing the grating parameters. For the purposes of comparison, Table 1 summarizes the performance of state-of-the-art chirped GMR filters reported in the literature. Ours is the first study to demonstrate sharp transmission dips with stable bandwidth and transmission intensity over a wide wavelength range based on the gradient grating period as well as the waveguide thickness. The bandwidth of the proposed gradient-thickness chirped GMR filter is superior to that of all state-of-the-art devices for TE as well as TM polarization.

 

Fig. 11 Measured optical transmission spectra of gradient-thickness chirped GMR filter under TE and TM polarized light at normal incidence. The GMR filter was filled with air (n = 1) or refractive index liquid (n = 1.8). The incident green and red laser beams had a spectral linewidth of less than 0.1 nm, which means that the transmission bandwidth of the measured transmission dips represents the ultimate spectral resolution of the gradient-thickness chirped GMR filter.

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Tables Icon

Table 1. Performances of state-of-the-art chirped GMR filters

Finally, we successfully applied the proposed chirped GMR filter as the dispersive device to reconstruct the optical spectrum of unknown light source by simply inverting the transmittance-versus-wavelength curves, which were obtained using the aforementioned third approach under TM-polarized light. For comparison, we also characterize the optical spectrum using a commercial optical spectrometer (USB4000 from Ocean Optics). Figure 12 shows the reconstructed optical spectra of a 6.88-mW white light emitting diode (LED) and a dual-peak laser beam combining from two different red laser sources (3.57-mW He-Ne laser and 3.74-mW diode laser). All reconstructed optical spectra are highly matched to the results obtained by a commercial optical spectrometer, indicating the feasibility of the proposed optical spectrometer based on continuously-chirped GMR filter. The zoom-in optical spectra in the insets of Fig. 12(b) further confirm the high spectral resolution of the chirped GMR filter, which is comparable or even better than the commercial optical spectrometer. Intensity fluctuation in dark wavelengths of reconstructed optical spectrum may be attributed to the slight variation in bandwidth and transmission intensity of chirped GMR filter along direction of interference.

 

Fig. 12 Measured optical spectra of (a) a 6.88-mW white light LED and (b) a laser beam combining from two different red laser sources (3.57-mW He-Ne laser and 3.74-mW diode laser) by a commercial optical spectrometer and the proposed optical spectrometer based on continuously-chirped GMR filter.

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4. Conclusions

In this study, we developed a tunable GMR filter based on continuously period-chirped (P = 270~400 nm) gratings using a high-index Ta2O5 waveguide layer with graded thickness (T = 126~162 nm). We defined the chirped grating structure using a convex mirror in Lloyd’s mirror interferometer to produce a light field with a progressively varying incident angle for interference. A novel masked e-beam evaporation process was also developed for the deposition of a Ta2O5 film with graded thickness. This resulted in a chirped GMR filter capable of providing sharp transmission dips at resonant wavelengths with a filter bandwidth of approximately 4.2 nm and 0.78 nm when respectively applied to TE and TM polarized light at normal incidence. It is crucial that dispersive devices in wavelength detection systems have consistent filter bandwidth and transmission intensity over a wide wavelength range. The proposed chirped GMR filter has a relatively constant waveguide-thickness-to-grating-period ratio of 0.4~0.46. As a result, all resonant wavelengths have similar transmission intensity of 0.18 and filter bandwidth of ~4.5 nm across the 200-nm operating wavelength. Unlike angular-resolved wavelength discrimination in prisms and diffraction gratings, wavelength discrimination in a chirped GMR filter is based on spatially resolved horizontal movement. This eliminates the need for wide spacing between the chirped GMR filter and detecting element to achieve high spectral resolution. Finally, we successfully applied the chirped GMR filter to loyally reconstruct the optical spectra of a white LED light source and a dual-peak laser beam. The proposed chirped GMR filter shows considerable promise as a dispersive device for applications in compact on-chip optical spectrometer and photonic lab-on-a-chip.

Funding

Ministry of Science and Technology, Taiwan (MOST) (MOST 104-2221-E-110-061, MOST 105-2221-E-110-075, and MOST 106-2221-E-110-060-MY3)

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20. C. Fang, B. Dai, Z. Li, A. Zahid, Q. Wang, B. Sheng, and D. Zhang, “Tunable guided-mode resonance filter with a gradient grating period fabricated by casting a stretched PDMS grating wedge,” Opt. Lett. 41(22), 5302–5305 (2016). [CrossRef]   [PubMed]  

21. D. W. Doobs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006). [CrossRef]  

22. Y.-J. Hung, H.-J. Chang, P.-C. Chang, J.-J. Lin, and T.-C. Kao, “Employing refractive beam shaping in a Lloyd’s interference lithography system for uniform periodic nanostructure formation,” J. Vac. Sci. Technol. B 35(3), 030601 (2017). [CrossRef]  

23. Y.-J. Hung, P.-C. Chang, Y.-N. Lin, and J.-J. Lin, “Compact mirror-tunable laser interference system for wafer-scale patterning of grating structures with flexible periodicity,” J. Vac. Sci. Technol. B 34(4), 040609 (2016). [CrossRef]  

24. K. Liu, H. Xu, H. Hu, Q. Gan, and A. N. Cartwright, “One-step fabrication of graded rainbow-colored holographic photopolymer reflection gratings,” Adv. Mater. 24(12), 1604–1609 (2012). [CrossRef]   [PubMed]  

25. A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. 13(4), 296–304 (1977). [CrossRef]  

26. A. Suzuki and K. Tada, “Fabrication of chirped gratings on GaAs optical waveguides,” Thin Solid Films 72(3), 419–426 (1980). [CrossRef]  

27. D. A. Bryan and J. K. Powers, “Improved holography for chirped gratings,” Opt. Lett. 5(9), 407–409 (1980). [CrossRef]   [PubMed]  

28. H. Kim, H. Jung, D.-H. Lee, K. B. Lee, and H. Jeon, “Period-chirped gratings fabricated by laser interference lithography with a concave Lloyd’s mirror,” Appl. Opt. 55(2), 354–359 (2016). [CrossRef]   [PubMed]  

29. R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008). [CrossRef]   [PubMed]  

30. T. H. Yoon, C. I. Eom, M. S. Chung, and H. J. Kong, “Diffractometric methods for absolute measurement of diffraction-grating spacings,” Opt. Lett. 24(2), 107–109 (1999). [CrossRef]   [PubMed]  

31. V. Korpelainen, A. Iho, J. Seppa, and A. Lassila, “High accuracy laser diffractometer: angle-scale traceability by the error separation method with a grating,” Meas. Sci. Technol. 20(8), 084020 (2009). [CrossRef]  

32. A. Bodere, D. Carpentier, A. Accard, and B. Fernier, “Grating fabrication and characterization method for wafers up to 2 in,” Mater. Sci. Eng. B 28(1-3), 293–295 (1994). [CrossRef]  

References

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  1. T. Fukazawa, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. 43(2), 646–647 (2004).
    [Crossref]
  2. K. Kodate and Y. Komai, “Compact spectroscopic sensor using an arrayed waveguide grating,” J. Opt. A, Pure Appl. Opt. 10(4), 044011 (2008).
    [Crossref]
  3. N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23(20), 1505–1507 (2011).
    [Crossref]
  4. S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
    [Crossref]
  5. X. Ma, M. Li, and J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photonics J. 5(2), 6600807 (2013).
    [Crossref]
  6. B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun. 282(15), 3168–3171 (2009).
    [Crossref]
  7. Z. Xia, A. A. Eftekhar, M. Soltani, B. Momeni, Q. Li, M. Chamanzar, S. Yegnanarayanan, and A. Adibi, “High resolution on-chip spectroscopy based on miniaturized microdonut resonators,” Opt. Express 19(13), 12356–12364 (2011).
    [Crossref] [PubMed]
  8. R. F. Wolffenbuttel, “State-of-the-art in integrated optical microspectrometers,” IEEE Trans. Instrum. Meas. 53(1), 197–202 (2004).
    [Crossref]
  9. S.-W. Wang, C. Xia, X. Chen, W. Lu, M. Li, H. Wang, W. Zheng, and T. Zhang, “Concept of a high-resolution miniature spectrometer using an integrated filter array,” Opt. Lett. 32(6), 632–634 (2007).
    [Crossref] [PubMed]
  10. R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
    [Crossref]
  11. Y. H. Ko and R. Magnusson, “Wideband dielectric metamaterial reflectors: Mie scattering or leaky Bloch mode resonance?” Optica 5(3), 289–294 (2018).
    [Crossref]
  12. M. J. Uddin and R. Magnusson, “Guided-mode resonant thermo-optic tunable filters,” IEEE Photonics Technol. Lett. 25(15), 1412–1415 (2013).
    [Crossref]
  13. S. Foland, K.-H. Choi, and J.-B. Lee, “Pressure-tunable guided-mode resonance sensor for single-wavelength characterization,” Opt. Lett. 35(23), 3871–3873 (2010).
    [Crossref] [PubMed]
  14. G. Xiao, Q. Zhu, Y. Shen, K. Li, M. Liu, Q. Zhuang, and C. Jin, “A tunable submicro-optofluidic polymer filter based on guided-mode resonance,” Nanoscale 7(8), 3429–3434 (2015).
    [Crossref] [PubMed]
  15. C.-T. Wang, H.-H. Hou, P.-C. Chang, C.-C. Li, H.-C. Jau, Y.-J. Hung, and T.-H. Lin, “Full-color reflectance-tunable filter based on liquid crystal cladded guided-mode resonant grating,” Opt. Express 24(20), 22892–22898 (2016).
    [Crossref] [PubMed]
  16. H.-A. Lin and C.-S. Huang, “Linear variable filter based on a gradient grating period guided-mode resonance filter,” IEEE Photonics Technol. Lett. 28, 1042–1045 (2016).
    [Crossref]
  17. G. J. Triggs, Y. Wang, C. P. Reardon, M. Fischer, G. J. O. Evans, and T. F. Krauss, “Chirped guided-mode resonance biosensor,” Optica 4(2), 229–234 (2017).
    [Crossref]
  18. H.-Y. Hsu, Y.-H. Lan, and C.-S. Huang, “A gradient grating period guided-mode resonance spectrometer,” IEEE Photonics J. 10(1), 4500109 (2018).
    [Crossref]
  19. E. Li, X. Chong, F. Ren, and A. X. Wang, “Broadband on-chip near-infrared spectroscopy based on a plasmonic grating filter array,” Opt. Lett. 41(9), 1913–1916 (2016).
    [Crossref] [PubMed]
  20. C. Fang, B. Dai, Z. Li, A. Zahid, Q. Wang, B. Sheng, and D. Zhang, “Tunable guided-mode resonance filter with a gradient grating period fabricated by casting a stretched PDMS grating wedge,” Opt. Lett. 41(22), 5302–5305 (2016).
    [Crossref] [PubMed]
  21. D. W. Doobs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006).
    [Crossref]
  22. Y.-J. Hung, H.-J. Chang, P.-C. Chang, J.-J. Lin, and T.-C. Kao, “Employing refractive beam shaping in a Lloyd’s interference lithography system for uniform periodic nanostructure formation,” J. Vac. Sci. Technol. B 35(3), 030601 (2017).
    [Crossref]
  23. Y.-J. Hung, P.-C. Chang, Y.-N. Lin, and J.-J. Lin, “Compact mirror-tunable laser interference system for wafer-scale patterning of grating structures with flexible periodicity,” J. Vac. Sci. Technol. B 34(4), 040609 (2016).
    [Crossref]
  24. K. Liu, H. Xu, H. Hu, Q. Gan, and A. N. Cartwright, “One-step fabrication of graded rainbow-colored holographic photopolymer reflection gratings,” Adv. Mater. 24(12), 1604–1609 (2012).
    [Crossref] [PubMed]
  25. A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. 13(4), 296–304 (1977).
    [Crossref]
  26. A. Suzuki and K. Tada, “Fabrication of chirped gratings on GaAs optical waveguides,” Thin Solid Films 72(3), 419–426 (1980).
    [Crossref]
  27. D. A. Bryan and J. K. Powers, “Improved holography for chirped gratings,” Opt. Lett. 5(9), 407–409 (1980).
    [Crossref] [PubMed]
  28. H. Kim, H. Jung, D.-H. Lee, K. B. Lee, and H. Jeon, “Period-chirped gratings fabricated by laser interference lithography with a concave Lloyd’s mirror,” Appl. Opt. 55(2), 354–359 (2016).
    [Crossref] [PubMed]
  29. R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008).
    [Crossref] [PubMed]
  30. T. H. Yoon, C. I. Eom, M. S. Chung, and H. J. Kong, “Diffractometric methods for absolute measurement of diffraction-grating spacings,” Opt. Lett. 24(2), 107–109 (1999).
    [Crossref] [PubMed]
  31. V. Korpelainen, A. Iho, J. Seppa, and A. Lassila, “High accuracy laser diffractometer: angle-scale traceability by the error separation method with a grating,” Meas. Sci. Technol. 20(8), 084020 (2009).
    [Crossref]
  32. A. Bodere, D. Carpentier, A. Accard, and B. Fernier, “Grating fabrication and characterization method for wafers up to 2 in,” Mater. Sci. Eng. B 28(1-3), 293–295 (1994).
    [Crossref]

2018 (2)

Y. H. Ko and R. Magnusson, “Wideband dielectric metamaterial reflectors: Mie scattering or leaky Bloch mode resonance?” Optica 5(3), 289–294 (2018).
[Crossref]

H.-Y. Hsu, Y.-H. Lan, and C.-S. Huang, “A gradient grating period guided-mode resonance spectrometer,” IEEE Photonics J. 10(1), 4500109 (2018).
[Crossref]

2017 (2)

Y.-J. Hung, H.-J. Chang, P.-C. Chang, J.-J. Lin, and T.-C. Kao, “Employing refractive beam shaping in a Lloyd’s interference lithography system for uniform periodic nanostructure formation,” J. Vac. Sci. Technol. B 35(3), 030601 (2017).
[Crossref]

G. J. Triggs, Y. Wang, C. P. Reardon, M. Fischer, G. J. O. Evans, and T. F. Krauss, “Chirped guided-mode resonance biosensor,” Optica 4(2), 229–234 (2017).
[Crossref]

2016 (6)

2015 (1)

G. Xiao, Q. Zhu, Y. Shen, K. Li, M. Liu, Q. Zhuang, and C. Jin, “A tunable submicro-optofluidic polymer filter based on guided-mode resonance,” Nanoscale 7(8), 3429–3434 (2015).
[Crossref] [PubMed]

2013 (2)

M. J. Uddin and R. Magnusson, “Guided-mode resonant thermo-optic tunable filters,” IEEE Photonics Technol. Lett. 25(15), 1412–1415 (2013).
[Crossref]

X. Ma, M. Li, and J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photonics J. 5(2), 6600807 (2013).
[Crossref]

2012 (1)

K. Liu, H. Xu, H. Hu, Q. Gan, and A. N. Cartwright, “One-step fabrication of graded rainbow-colored holographic photopolymer reflection gratings,” Adv. Mater. 24(12), 1604–1609 (2012).
[Crossref] [PubMed]

2011 (2)

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23(20), 1505–1507 (2011).
[Crossref]

Z. Xia, A. A. Eftekhar, M. Soltani, B. Momeni, Q. Li, M. Chamanzar, S. Yegnanarayanan, and A. Adibi, “High resolution on-chip spectroscopy based on miniaturized microdonut resonators,” Opt. Express 19(13), 12356–12364 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (2)

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun. 282(15), 3168–3171 (2009).
[Crossref]

V. Korpelainen, A. Iho, J. Seppa, and A. Lassila, “High accuracy laser diffractometer: angle-scale traceability by the error separation method with a grating,” Meas. Sci. Technol. 20(8), 084020 (2009).
[Crossref]

2008 (2)

R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008).
[Crossref] [PubMed]

K. Kodate and Y. Komai, “Compact spectroscopic sensor using an arrayed waveguide grating,” J. Opt. A, Pure Appl. Opt. 10(4), 044011 (2008).
[Crossref]

2007 (1)

2006 (1)

D. W. Doobs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006).
[Crossref]

2004 (3)

T. Fukazawa, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. 43(2), 646–647 (2004).
[Crossref]

R. F. Wolffenbuttel, “State-of-the-art in integrated optical microspectrometers,” IEEE Trans. Instrum. Meas. 53(1), 197–202 (2004).
[Crossref]

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
[Crossref]

1999 (1)

1994 (1)

A. Bodere, D. Carpentier, A. Accard, and B. Fernier, “Grating fabrication and characterization method for wafers up to 2 in,” Mater. Sci. Eng. B 28(1-3), 293–295 (1994).
[Crossref]

1992 (1)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

1980 (2)

D. A. Bryan and J. K. Powers, “Improved holography for chirped gratings,” Opt. Lett. 5(9), 407–409 (1980).
[Crossref] [PubMed]

A. Suzuki and K. Tada, “Fabrication of chirped gratings on GaAs optical waveguides,” Thin Solid Films 72(3), 419–426 (1980).
[Crossref]

1977 (1)

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. 13(4), 296–304 (1977).
[Crossref]

Accard, A.

A. Bodere, D. Carpentier, A. Accard, and B. Fernier, “Grating fabrication and characterization method for wafers up to 2 in,” Mater. Sci. Eng. B 28(1-3), 293–295 (1994).
[Crossref]

Adibi, A.

Z. Xia, A. A. Eftekhar, M. Soltani, B. Momeni, Q. Li, M. Chamanzar, S. Yegnanarayanan, and A. Adibi, “High resolution on-chip spectroscopy based on miniaturized microdonut resonators,” Opt. Express 19(13), 12356–12364 (2011).
[Crossref] [PubMed]

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun. 282(15), 3168–3171 (2009).
[Crossref]

Askari, M.

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun. 282(15), 3168–3171 (2009).
[Crossref]

Baba, T.

T. Fukazawa, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. 43(2), 646–647 (2004).
[Crossref]

Baets, R.

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23(20), 1505–1507 (2011).
[Crossref]

Balakrishnan, A.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
[Crossref]

Bodere, A.

A. Bodere, D. Carpentier, A. Accard, and B. Fernier, “Grating fabrication and characterization method for wafers up to 2 in,” Mater. Sci. Eng. B 28(1-3), 293–295 (1994).
[Crossref]

Bogaerts, W.

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23(20), 1505–1507 (2011).
[Crossref]

Bryan, D. A.

Carpentier, D.

A. Bodere, D. Carpentier, A. Accard, and B. Fernier, “Grating fabrication and characterization method for wafers up to 2 in,” Mater. Sci. Eng. B 28(1-3), 293–295 (1994).
[Crossref]

Cartwright, A. N.

K. Liu, H. Xu, H. Hu, Q. Gan, and A. N. Cartwright, “One-step fabrication of graded rainbow-colored holographic photopolymer reflection gratings,” Adv. Mater. 24(12), 1604–1609 (2012).
[Crossref] [PubMed]

Chamanzar, M.

Chang, H.-J.

Y.-J. Hung, H.-J. Chang, P.-C. Chang, J.-J. Lin, and T.-C. Kao, “Employing refractive beam shaping in a Lloyd’s interference lithography system for uniform periodic nanostructure formation,” J. Vac. Sci. Technol. B 35(3), 030601 (2017).
[Crossref]

Chang, P.-C.

Y.-J. Hung, H.-J. Chang, P.-C. Chang, J.-J. Lin, and T.-C. Kao, “Employing refractive beam shaping in a Lloyd’s interference lithography system for uniform periodic nanostructure formation,” J. Vac. Sci. Technol. B 35(3), 030601 (2017).
[Crossref]

Y.-J. Hung, P.-C. Chang, Y.-N. Lin, and J.-J. Lin, “Compact mirror-tunable laser interference system for wafer-scale patterning of grating structures with flexible periodicity,” J. Vac. Sci. Technol. B 34(4), 040609 (2016).
[Crossref]

C.-T. Wang, H.-H. Hou, P.-C. Chang, C.-C. Li, H.-C. Jau, Y.-J. Hung, and T.-H. Lin, “Full-color reflectance-tunable filter based on liquid crystal cladded guided-mode resonant grating,” Opt. Express 24(20), 22892–22898 (2016).
[Crossref] [PubMed]

Charbonneau, S.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
[Crossref]

Cheben, P.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
[Crossref]

Chen, X.

Choi, K.-H.

Chong, X.

Chung, M. S.

Cloutier, M.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
[Crossref]

Cunningham, B. T.

D. W. Doobs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006).
[Crossref]

Dai, B.

Doobs, D. W.

D. W. Doobs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006).
[Crossref]

Dossou, K.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
[Crossref]

Eftekhar, A. A.

Eom, C. I.

Erickson, L.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
[Crossref]

Evans, G. J. O.

Fang, C.

Fernier, B.

A. Bodere, D. Carpentier, A. Accard, and B. Fernier, “Grating fabrication and characterization method for wafers up to 2 in,” Mater. Sci. Eng. B 28(1-3), 293–295 (1994).
[Crossref]

Fischer, M.

Foland, S.

Fukazawa, T.

T. Fukazawa, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. 43(2), 646–647 (2004).
[Crossref]

Gan, Q.

K. Liu, H. Xu, H. Hu, Q. Gan, and A. N. Cartwright, “One-step fabrication of graded rainbow-colored holographic photopolymer reflection gratings,” Adv. Mater. 24(12), 1604–1609 (2012).
[Crossref] [PubMed]

Gao, M.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
[Crossref]

Gershkovich, I.

D. W. Doobs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006).
[Crossref]

He, J.

X. Ma, M. Li, and J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photonics J. 5(2), 6600807 (2013).
[Crossref]

Hens, Z.

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23(20), 1505–1507 (2011).
[Crossref]

Hosseini, E. S.

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun. 282(15), 3168–3171 (2009).
[Crossref]

Hou, H.-H.

Hsu, H.-Y.

H.-Y. Hsu, Y.-H. Lan, and C.-S. Huang, “A gradient grating period guided-mode resonance spectrometer,” IEEE Photonics J. 10(1), 4500109 (2018).
[Crossref]

Hu, H.

K. Liu, H. Xu, H. Hu, Q. Gan, and A. N. Cartwright, “One-step fabrication of graded rainbow-colored holographic photopolymer reflection gratings,” Adv. Mater. 24(12), 1604–1609 (2012).
[Crossref] [PubMed]

Huang, C.-S.

H.-Y. Hsu, Y.-H. Lan, and C.-S. Huang, “A gradient grating period guided-mode resonance spectrometer,” IEEE Photonics J. 10(1), 4500109 (2018).
[Crossref]

H.-A. Lin and C.-S. Huang, “Linear variable filter based on a gradient grating period guided-mode resonance filter,” IEEE Photonics Technol. Lett. 28, 1042–1045 (2016).
[Crossref]

Hung, Y.-J.

Y.-J. Hung, H.-J. Chang, P.-C. Chang, J.-J. Lin, and T.-C. Kao, “Employing refractive beam shaping in a Lloyd’s interference lithography system for uniform periodic nanostructure formation,” J. Vac. Sci. Technol. B 35(3), 030601 (2017).
[Crossref]

Y.-J. Hung, P.-C. Chang, Y.-N. Lin, and J.-J. Lin, “Compact mirror-tunable laser interference system for wafer-scale patterning of grating structures with flexible periodicity,” J. Vac. Sci. Technol. B 34(4), 040609 (2016).
[Crossref]

C.-T. Wang, H.-H. Hou, P.-C. Chang, C.-C. Li, H.-C. Jau, Y.-J. Hung, and T.-H. Lin, “Full-color reflectance-tunable filter based on liquid crystal cladded guided-mode resonant grating,” Opt. Express 24(20), 22892–22898 (2016).
[Crossref] [PubMed]

Iho, A.

V. Korpelainen, A. Iho, J. Seppa, and A. Lassila, “High accuracy laser diffractometer: angle-scale traceability by the error separation method with a grating,” Meas. Sci. Technol. 20(8), 084020 (2009).
[Crossref]

Janz, S.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
[Crossref]

Jau, H.-C.

Jeon, H.

Jin, C.

G. Xiao, Q. Zhu, Y. Shen, K. Li, M. Liu, Q. Zhuang, and C. Jin, “A tunable submicro-optofluidic polymer filter based on guided-mode resonance,” Nanoscale 7(8), 3429–3434 (2015).
[Crossref] [PubMed]

Jung, H.

Kao, T.-C.

Y.-J. Hung, H.-J. Chang, P.-C. Chang, J.-J. Lin, and T.-C. Kao, “Employing refractive beam shaping in a Lloyd’s interference lithography system for uniform periodic nanostructure formation,” J. Vac. Sci. Technol. B 35(3), 030601 (2017).
[Crossref]

Katzir, A.

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. 13(4), 296–304 (1977).
[Crossref]

Kim, H.

Ko, Y. H.

Kodate, K.

K. Kodate and Y. Komai, “Compact spectroscopic sensor using an arrayed waveguide grating,” J. Opt. A, Pure Appl. Opt. 10(4), 044011 (2008).
[Crossref]

Komai, Y.

K. Kodate and Y. Komai, “Compact spectroscopic sensor using an arrayed waveguide grating,” J. Opt. A, Pure Appl. Opt. 10(4), 044011 (2008).
[Crossref]

Kong, H. J.

Korpelainen, V.

V. Korpelainen, A. Iho, J. Seppa, and A. Lassila, “High accuracy laser diffractometer: angle-scale traceability by the error separation method with a grating,” Meas. Sci. Technol. 20(8), 084020 (2009).
[Crossref]

Krauss, T. F.

Krug, P. A.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
[Crossref]

Lan, Y.-H.

H.-Y. Hsu, Y.-H. Lan, and C.-S. Huang, “A gradient grating period guided-mode resonance spectrometer,” IEEE Photonics J. 10(1), 4500109 (2018).
[Crossref]

Lassila, A.

V. Korpelainen, A. Iho, J. Seppa, and A. Lassila, “High accuracy laser diffractometer: angle-scale traceability by the error separation method with a grating,” Meas. Sci. Technol. 20(8), 084020 (2009).
[Crossref]

Lee, D.-H.

Lee, J.-B.

Lee, K. B.

Li, C.-C.

Li, E.

Li, K.

G. Xiao, Q. Zhu, Y. Shen, K. Li, M. Liu, Q. Zhuang, and C. Jin, “A tunable submicro-optofluidic polymer filter based on guided-mode resonance,” Nanoscale 7(8), 3429–3434 (2015).
[Crossref] [PubMed]

Li, M.

X. Ma, M. Li, and J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photonics J. 5(2), 6600807 (2013).
[Crossref]

S.-W. Wang, C. Xia, X. Chen, W. Lu, M. Li, H. Wang, W. Zheng, and T. Zhang, “Concept of a high-resolution miniature spectrometer using an integrated filter array,” Opt. Lett. 32(6), 632–634 (2007).
[Crossref] [PubMed]

Li, Q.

Li, Z.

Lin, H.-A.

H.-A. Lin and C.-S. Huang, “Linear variable filter based on a gradient grating period guided-mode resonance filter,” IEEE Photonics Technol. Lett. 28, 1042–1045 (2016).
[Crossref]

Lin, J.-J.

Y.-J. Hung, H.-J. Chang, P.-C. Chang, J.-J. Lin, and T.-C. Kao, “Employing refractive beam shaping in a Lloyd’s interference lithography system for uniform periodic nanostructure formation,” J. Vac. Sci. Technol. B 35(3), 030601 (2017).
[Crossref]

Y.-J. Hung, P.-C. Chang, Y.-N. Lin, and J.-J. Lin, “Compact mirror-tunable laser interference system for wafer-scale patterning of grating structures with flexible periodicity,” J. Vac. Sci. Technol. B 34(4), 040609 (2016).
[Crossref]

Lin, T.-H.

Lin, Y.-N.

Y.-J. Hung, P.-C. Chang, Y.-N. Lin, and J.-J. Lin, “Compact mirror-tunable laser interference system for wafer-scale patterning of grating structures with flexible periodicity,” J. Vac. Sci. Technol. B 34(4), 040609 (2016).
[Crossref]

Liu, K.

K. Liu, H. Xu, H. Hu, Q. Gan, and A. N. Cartwright, “One-step fabrication of graded rainbow-colored holographic photopolymer reflection gratings,” Adv. Mater. 24(12), 1604–1609 (2012).
[Crossref] [PubMed]

Liu, M.

G. Xiao, Q. Zhu, Y. Shen, K. Li, M. Liu, Q. Zhuang, and C. Jin, “A tunable submicro-optofluidic polymer filter based on guided-mode resonance,” Nanoscale 7(8), 3429–3434 (2015).
[Crossref] [PubMed]

Livanos, A. C.

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. 13(4), 296–304 (1977).
[Crossref]

Lu, W.

Ma, X.

X. Ma, M. Li, and J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photonics J. 5(2), 6600807 (2013).
[Crossref]

Magnusson, R.

Y. H. Ko and R. Magnusson, “Wideband dielectric metamaterial reflectors: Mie scattering or leaky Bloch mode resonance?” Optica 5(3), 289–294 (2018).
[Crossref]

M. J. Uddin and R. Magnusson, “Guided-mode resonant thermo-optic tunable filters,” IEEE Photonics Technol. Lett. 25(15), 1412–1415 (2013).
[Crossref]

R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008).
[Crossref] [PubMed]

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

Momeni, B.

Z. Xia, A. A. Eftekhar, M. Soltani, B. Momeni, Q. Li, M. Chamanzar, S. Yegnanarayanan, and A. Adibi, “High resolution on-chip spectroscopy based on miniaturized microdonut resonators,” Opt. Express 19(13), 12356–12364 (2011).
[Crossref] [PubMed]

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun. 282(15), 3168–3171 (2009).
[Crossref]

Ohno, F.

T. Fukazawa, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. 43(2), 646–647 (2004).
[Crossref]

Powers, J. K.

Reardon, C. P.

Ren, F.

Seppa, J.

V. Korpelainen, A. Iho, J. Seppa, and A. Lassila, “High accuracy laser diffractometer: angle-scale traceability by the error separation method with a grating,” Meas. Sci. Technol. 20(8), 084020 (2009).
[Crossref]

Shellan, J. B.

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. 13(4), 296–304 (1977).
[Crossref]

Shen, Y.

G. Xiao, Q. Zhu, Y. Shen, K. Li, M. Liu, Q. Zhuang, and C. Jin, “A tunable submicro-optofluidic polymer filter based on guided-mode resonance,” Nanoscale 7(8), 3429–3434 (2015).
[Crossref] [PubMed]

Sheng, B.

Shokooh-Saremi, M.

Soltani, M.

Z. Xia, A. A. Eftekhar, M. Soltani, B. Momeni, Q. Li, M. Chamanzar, S. Yegnanarayanan, and A. Adibi, “High resolution on-chip spectroscopy based on miniaturized microdonut resonators,” Opt. Express 19(13), 12356–12364 (2011).
[Crossref] [PubMed]

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun. 282(15), 3168–3171 (2009).
[Crossref]

Suzuki, A.

A. Suzuki and K. Tada, “Fabrication of chirped gratings on GaAs optical waveguides,” Thin Solid Films 72(3), 419–426 (1980).
[Crossref]

Tada, K.

A. Suzuki and K. Tada, “Fabrication of chirped gratings on GaAs optical waveguides,” Thin Solid Films 72(3), 419–426 (1980).
[Crossref]

Triggs, G. J.

Uddin, M. J.

M. J. Uddin and R. Magnusson, “Guided-mode resonant thermo-optic tunable filters,” IEEE Photonics Technol. Lett. 25(15), 1412–1415 (2013).
[Crossref]

Wang, A. X.

Wang, C.-T.

Wang, H.

Wang, Q.

Wang, S. S.

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

Wang, S.-W.

Wang, Y.

Wolffenbuttel, R. F.

R. F. Wolffenbuttel, “State-of-the-art in integrated optical microspectrometers,” IEEE Trans. Instrum. Meas. 53(1), 197–202 (2004).
[Crossref]

Xia, C.

Xia, Z.

Xiao, G.

G. Xiao, Q. Zhu, Y. Shen, K. Li, M. Liu, Q. Zhuang, and C. Jin, “A tunable submicro-optofluidic polymer filter based on guided-mode resonance,” Nanoscale 7(8), 3429–3434 (2015).
[Crossref] [PubMed]

Xu, H.

K. Liu, H. Xu, H. Hu, Q. Gan, and A. N. Cartwright, “One-step fabrication of graded rainbow-colored holographic photopolymer reflection gratings,” Adv. Mater. 24(12), 1604–1609 (2012).
[Crossref] [PubMed]

Yariv, A.

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. 13(4), 296–304 (1977).
[Crossref]

Yebo, N. A.

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23(20), 1505–1507 (2011).
[Crossref]

Yegnanarayanan, S.

Yoon, T. H.

Zahid, A.

Zhang, D.

Zhang, T.

Zheng, W.

Zhu, Q.

G. Xiao, Q. Zhu, Y. Shen, K. Li, M. Liu, Q. Zhuang, and C. Jin, “A tunable submicro-optofluidic polymer filter based on guided-mode resonance,” Nanoscale 7(8), 3429–3434 (2015).
[Crossref] [PubMed]

Zhuang, Q.

G. Xiao, Q. Zhu, Y. Shen, K. Li, M. Liu, Q. Zhuang, and C. Jin, “A tunable submicro-optofluidic polymer filter based on guided-mode resonance,” Nanoscale 7(8), 3429–3434 (2015).
[Crossref] [PubMed]

Adv. Mater. (1)

K. Liu, H. Xu, H. Hu, Q. Gan, and A. N. Cartwright, “One-step fabrication of graded rainbow-colored holographic photopolymer reflection gratings,” Adv. Mater. 24(12), 1604–1609 (2012).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

D. W. Doobs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006).
[Crossref]

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, “Chirped gratings in integrated optics,” IEEE J. Quantum Electron. 13(4), 296–304 (1977).
[Crossref]

IEEE Photonics J. (2)

H.-Y. Hsu, Y.-H. Lan, and C.-S. Huang, “A gradient grating period guided-mode resonance spectrometer,” IEEE Photonics J. 10(1), 4500109 (2018).
[Crossref]

X. Ma, M. Li, and J. He, “CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array,” IEEE Photonics J. 5(2), 6600807 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (4)

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23(20), 1505–1507 (2011).
[Crossref]

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, K. Dossou, L. Erickson, M. Gao, and P. A. Krug, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16(2), 503–505 (2004).
[Crossref]

M. J. Uddin and R. Magnusson, “Guided-mode resonant thermo-optic tunable filters,” IEEE Photonics Technol. Lett. 25(15), 1412–1415 (2013).
[Crossref]

H.-A. Lin and C.-S. Huang, “Linear variable filter based on a gradient grating period guided-mode resonance filter,” IEEE Photonics Technol. Lett. 28, 1042–1045 (2016).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

R. F. Wolffenbuttel, “State-of-the-art in integrated optical microspectrometers,” IEEE Trans. Instrum. Meas. 53(1), 197–202 (2004).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

K. Kodate and Y. Komai, “Compact spectroscopic sensor using an arrayed waveguide grating,” J. Opt. A, Pure Appl. Opt. 10(4), 044011 (2008).
[Crossref]

J. Vac. Sci. Technol. B (2)

Y.-J. Hung, H.-J. Chang, P.-C. Chang, J.-J. Lin, and T.-C. Kao, “Employing refractive beam shaping in a Lloyd’s interference lithography system for uniform periodic nanostructure formation,” J. Vac. Sci. Technol. B 35(3), 030601 (2017).
[Crossref]

Y.-J. Hung, P.-C. Chang, Y.-N. Lin, and J.-J. Lin, “Compact mirror-tunable laser interference system for wafer-scale patterning of grating structures with flexible periodicity,” J. Vac. Sci. Technol. B 34(4), 040609 (2016).
[Crossref]

Jpn. J. Appl. Phys. (1)

T. Fukazawa, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. 43(2), 646–647 (2004).
[Crossref]

Mater. Sci. Eng. B (1)

A. Bodere, D. Carpentier, A. Accard, and B. Fernier, “Grating fabrication and characterization method for wafers up to 2 in,” Mater. Sci. Eng. B 28(1-3), 293–295 (1994).
[Crossref]

Meas. Sci. Technol. (1)

V. Korpelainen, A. Iho, J. Seppa, and A. Lassila, “High accuracy laser diffractometer: angle-scale traceability by the error separation method with a grating,” Meas. Sci. Technol. 20(8), 084020 (2009).
[Crossref]

Nanoscale (1)

G. Xiao, Q. Zhu, Y. Shen, K. Li, M. Liu, Q. Zhuang, and C. Jin, “A tunable submicro-optofluidic polymer filter based on guided-mode resonance,” Nanoscale 7(8), 3429–3434 (2015).
[Crossref] [PubMed]

Opt. Commun. (1)

B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun. 282(15), 3168–3171 (2009).
[Crossref]

Opt. Express (3)

Opt. Lett. (6)

Optica (2)

Thin Solid Films (1)

A. Suzuki and K. Tada, “Fabrication of chirped gratings on GaAs optical waveguides,” Thin Solid Films 72(3), 419–426 (1980).
[Crossref]

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Figures (12)

Fig. 1
Fig. 1 Schematic diagram showing continuously-chirped Ta2O5 guided mode resonance filter. Gradient variation in the grating period is crucial to creation of a compact high-resolution optical spectrometer.
Fig. 2
Fig. 2 (a,b) Simulated optical transmission spectra of continuously-chirped air-cladded Ta2O5 based GMR filter under normal incidence TE- and TM-polarized light, respectively. (c,d) Simulated optical transmission spectra of chirped GMR filter filled with refractive index liquid for TE polarization, as well as a Ta2O5 waveguide layer with (c) fixed and (d) graded thicknesses.
Fig. 3
Fig. 3 (a) Schematic illustration of one-beam Lloyd’s laser interferometer equipped with a convex mirror. Physical parameters x0, x1, y0, y1, and R are graphically defined in the figure. (b) Chirped grating periods as a function of interference position y calculated for convex Lloyd’s mirrors with three different curvature radii: R = 193, 500, and 1000 mm.
Fig. 4
Fig. 4 Schematic illustration of proposed LIL system used for chirped grating fabrication.
Fig. 5
Fig. 5 Grating period versus interference position using direct laser beams with various incident angles θ1. Solid lines indicate theoretical predictions, whereas hollow symbols indicate experimental data.
Fig. 6
Fig. 6 Mapping of (a) periodicity and (b) diffraction efficiency of chirped gratings over 2-inch wafer area using custom-made optical diffractometer.
Fig. 7
Fig. 7 Cross-sectional SEM views of chirped GMR filter in positions where grating period is (a) 277 nm and (b) 394 nm; (c) thickness trends of Ta2O5 waveguide layer in two as-fabricated chirped GMR filters (red rectangles indicate optimal thickness range for given grating period, based on RCWA simulations).
Fig. 8
Fig. 8 (a) Photograph of chirped GMR filter illuminated by white light source to reveal rainbow-colored diffraction image; (b) experiment setup used to characterize spectral resolution of chirped GMR filter.
Fig. 9
Fig. 9 Chirped GMR filter with a fixed waveguide layer: (a) Measured optical transmission spectra of chirped GMR filter and (b) corresponding transmission bandwidths and intensities along direction of interference for TE polarization (thickness of Ta2O5 waveguide layer fixed at 110 nm).
Fig. 10
Fig. 10 Chirped GMR filter with a graded waveguide layer: (a) Measured optical transmission spectra of chirped GMR filter and (b) corresponding transmission bandwidths and intensities along direction of interference for TE polarization (thickness of Ta2O5 waveguide layer 126~162 nm).
Fig. 11
Fig. 11 Measured optical transmission spectra of gradient-thickness chirped GMR filter under TE and TM polarized light at normal incidence. The GMR filter was filled with air (n = 1) or refractive index liquid (n = 1.8). The incident green and red laser beams had a spectral linewidth of less than 0.1 nm, which means that the transmission bandwidth of the measured transmission dips represents the ultimate spectral resolution of the gradient-thickness chirped GMR filter.
Fig. 12
Fig. 12 Measured optical spectra of (a) a 6.88-mW white light LED and (b) a laser beam combining from two different red laser sources (3.57-mW He-Ne laser and 3.74-mW diode laser) by a commercial optical spectrometer and the proposed optical spectrometer based on continuously-chirped GMR filter.

Tables (1)

Tables Icon

Table 1 Performances of state-of-the-art chirped GMR filters

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

Λ= λ uv sin θ 1 +sin θ 2 = λ uv sin θ 1 +sin[ θ 1 +2 sin 1 ( x/R ) ]
φ=2 sin 1 ( x/R )
y( φ )=( Rsinφ+ x 0 + x 1 )tan( θ 1 +2φ ) y 1 y 0 y 1 =( x 0 + x 1 )tan θ 1 y 0 =R( 1cosφ )

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